I am currently working through an old qualifying exam problem:
Calculate the fundamental group and homology groups of the space $X$ obtained from the union $S^2\cup T$ where $S^2$ is the 2-sphere and $T$ is the torus by identifying 2 disjoint circles on the sphere to 2 disjoint circles on the torus. (Picture half of the torus being inside the sphere and half outside)
As for the fundamental group, I know that I can't use Seifert-Van Kampen on $A=S^2$ and $B=T$ because $A\cap B$ is 2 disjoint circles and hence is not path connected.
Without Seifert-Van Kampen, I am not sure how to approach the problem. I would appreciate any hints or outlines on how to proceed.